Hello readers. Before reading this, know that I do not in any way think everything I am about to write is correct, or true, or even half way sensible. It is simply a collection of thoughts that, in my eyes, makes some sense. Anyway.
About five minutes ago I was lying in bed attempting to nap, and whilst considering the implications of the Heisenberg uncertainty principle, I had two interesting considerations.
The first is that I do not believe doubling in mathematics is the opposite to halving. This led me to the conclusion and belief that I can argue, to a convincing degree, the flawed nature of the concept of infinity and that it is in fact inherently ‘wrong’.
First, let us consider the nature of halving. Any arbitrary figure you consider can be halved. The number you receive as the output of this operation can also be halved. This process of halving, or dividing a number by two, and then taking the output and repeating the process, can be done as many times as one would like. But no matter how many times you half a number, conceptually at least, it never reaches zero.
Then, consider doubling. As with halving, any number can be doubled, or multiplied by two, and it’s output doubled, any number of times. But it does not, as with halving, have a defined limit which it can never reach. It in fact, continues forever, you can double a number ‘infinitely’.
The comparison of the two outcomes of indefinite repetition of the processes combined with knowledge of the basic laws of mathematics leads me to conclude that they are not opposites of each other.
When one considers the outcome of the repetition of halving, there is a degree of uncertainty as to the actual outcome. To some extent, it can be defined; You can know roughly the outcome of the operation. Defined as, for example: 0 < x < 1. We know that the outcome will be less than one, as no matter what number is chosen, eventually the product becomes less than one. We also know that since any number, no matter how insignificantly small, can be halved and the product still have a value, it can never reach 0.
However we know from simply taking an example from another area of mathematics, calculus, that we can define the outcome further. When considering the gradient of a line that is constantly changing, we make an assumption. This assumption is that when we consider two points on the line extremely close to each other, the difference in the gradient is so minute that in mathematical terms, it can be equated to zero. This assumption is a fundamental consideration for differentiation and intergration, and can be applied to the process of halving above.
Thus, we can conclude that in fact, halving a number eventually does lead to the output 0. Eventually, the product of the operation becomes so small, that in mathematical terms is is equivalent to zero. Something similar is observed when considering that 0.9 recurring is equal to 1. There, we have defined the outcome of the operation.
The difference is obvious when considering the outcome of the operation of doubling. We can not accurately define the outcome of indefinite doubling, it simple goes on forever. We define this ‘going on forever’ as infinity. This basic difference, the fact that there is no definite outcome for the operation of doubling, leads me to conclude that they are not in fact opposite operations.
This lead me to wonder if 0 was in fact a finite outcome. Was 0 not the absence of something, but in fact the presence of infinite nothing.
I decided the best way to approach this question was from a practical point of view. Consider a 100 meter sprint. When measuring the distance of this race, it can be defined in two ways: The distance left to cover, and the distance covered. My proof for the conclusion that 0 is in fact a finite outcome is the fact in this case we can clearly measure it. Before the race has started, no one has travelled any distance, and you can measure this; the position they are in is measurably symmetrical as the position they were in. They have not travelled any distance, there is a lack of something to measure (rather than they have travelled an infinite lack of distance, or there is an infinite amount of nothing to measure.).
Satisfied with my conclusion that 0 is a finite article, I began to wonder if infinity is actually finite. Again, I took a practical consideration, but on a larger scale; the number of particles in the Universe. You might think this is infinite, but in fact it is not. There are a limited and measurable amount (about 10^80) amount of particles in the universe. Since, in this respect, the Universe is limited, surely it is ridiculous to consider any concept of infinity ?
Agreed, this is a very simple argument, but a more complex and indeed convincing one based on the same principle is the case of the infinite monkey theorem: http://en.wikipedia.org/wiki/Infinite_monkey_theorem.
Essentially, my argument is that all mathematics and mathematical proofs can be related in some form to the physical Universe. Infinity, however, can not. Therefore it has no place in mathematics and indeed no place at all anywhere. Things can be very very large, or indeed very very small, but they can not be infinite or indefinite.
Anyways, that is enough for one day. I have thought of several arguments against my conclusions, but I will explore those another day.
Thanks for reading.
- keepingthingssimple posted this